Error analysis of staggered finite difference finite volume schemes on unstructured meshes

TL;DR

This paper analyzes error estimates for staggered finite difference finite volume schemes on unstructured meshes, demonstrating first-order convergence and improvements under specific mesh conditions for elliptic and Stokes problems.

Contribution

It introduces a combined approach for error analysis that applies to unstructured meshes, achieving first-order convergence and higher rates under certain mesh conditions.

Findings

First-order convergence on unstructured meshes for elliptic problems.

Improved convergence rates on meshes with special properties.

Extension of error analysis to the 2D Stokes problem.

Abstract

This work combines the consistency in lower-order differential operators with external approximations of functional spaces to obtain error estimates for finite difference finite volume schemes on unstructured non-uniform meshes. This combined approach is first applied to the one-dimensional elliptic boundary value problem on non-uniform meshes, and a first-order convergence rate is obtained, which agrees with the results previously reported. The approach is also applied to the staggered MAC scheme for the two-dimensional incompressible Stokes problem on unstructured meshes. A first-order convergence rate is obtained, which improves over a previously reported result in that it also holds on unstructured meshes. For both problems considered in this work, the convergence rate is one order higher on meshes satisfying special requirements.